Combined_Doc

AN APPLICATION OF THE VIKOR METHOD
TO THE SELECTION OF POWER
GENERATION METHOD
ADEYEMO, AFEEZ TOMILADE
Matriculation Number: 134287
A project work submitted to the Department of Industrial and
Production Engineering, Faculty of Technology, University of Ibadan
in Partial Fulfilment of the Requirements for the Award of the
Bachelor of Science (B.Sc.) Degree in Industrial and Production
Engineering.
October 2011

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CERTIFICATION
This is to certify that Adeyemo, Afeez Tomilade carried out this study in the Department of
Industrial and Production Engineering, Faculty of Technology, University of Ibadan, Nigeria.
--------------------------------- ---------------------------
Project Supervisor Date
Dr. A.D. Adeyeye
B.Sc. (Ife), M.Sc. (Ibadan), Ph.D (Ibadan)
-------------------------------- -------------------------------
Head of Department Date
Dr. V.O. Oladokun
B.Sc. (Ife), M.Sc. (Ibadan), Ph.D (Ibadan)

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DEDICATION
There is nothing more heart-warming than the assurance of good friends and family to love
and care for one another, and the stronger these ties the harder when such relationships finally
come to an end. With the deepest love, I dedicate my project to my very dear parents; to the
departed and the one who remains still…
And last, but certainly not the least, to Allaah who fashions all according to His will. Indeed,
all praise and thanks are due to none but Him.

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ACKNOWLEDGEMENT
While this work could not have been completed without many days, weeks, and months of
independent work and studies, even less would it have been possible without the steady
support and encouragement of many others to whom I wish to express my sincerest
appreciation in these few words of recognition.
First of all, I am very grateful to my project supervisor Dr. David Adeyeye for his continued
support and guidance throughout this work. Working with him was a constant source of
challenging and at the same time fulfilling experiences and I greatly appreciate and value his
expertise of which he was always willing to share. Moreover, I am especially thankful to him
for his comments and corrections of earlier versions of this manuscript.
While too numerous to mention, I am indebted to many other teachers and faculty members
who prepared me for this work and who share important contributions to my overall
advancement as a student and person. Only in most recent matters, I extend my sincere
thanks to Professor Charles-Owaba, Dr Victor Oladokun, Dr. Osita Anyaeche and Dr.
Kolawole. I would also like to thank the departmental administration and its staff for their
general assistance to me over the years.
I will not attempt to list all of my friends at U.I. who have made the past several years so
much fun . . . to do so would take up far too much room. Top of the list is my namesake and
roommate of three years, Afeez Alebiosu. Others like Saheed, Sir T and Mubarak have made
the U.I. experience more worthwhile. It has been rewarding knowing you all. Thanks!
Finally, it is beyond doubt that any other recognition falls behind the deep gratefulness that I
owe to my family, mostly especially my mother. Her caring love and unflinching support has
been my steady source of joy, and to make her proud is ample motivation for always moving
on.

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ABSTRACT
The availability of electricity at reasonable prices is most essential for the continued growth
and development of any industrial setup. In the absence of a reliable source in the public grid,
most organisations are left to make a decision to make private provision. This decision
involves a number of factors to be considered and has a major impact on the success of such
organisations.
This work describes a model of a fuzzy decision support system in multi-criteria analysis
approach for selecting the best alternative(s) in self-generation of power for an industrial
facility. 8 alternative means of generation were evaluated with regard to 5 different decision
criteria. The Fuzzy Analytic Hierarchy Process (FAHP) method is used to determine the
preference weightings of criteria for decision makers by subjective perception (natural
language). A fuzzy approach using triangular fuzzy numbers was adopted to approximate the
human subjective evaluation process. The actual selection process was carried out using the
VIKOR model which is an upshot of the Compromise Programming algorithm.
The model determined that the gas turbine, gas internal combustion engine, fuel cell,
microturbine and bi-fuel engine were the compromise set of alternatives due. As a result, one
or a combination of these alternatives serve as the best option for generating power for an
industrial facility in Nigeria.

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TABLE OF CONTENTS
CHAPTER ONE: INTRODUCTION........................................................................................1
1.1 Overview.........................................................................................................................1
1.2 Situation of Generation and Distribution of Electricity in Nigeria.................................1
1.3 Problem Statement..........................................................................................................3
1.4 Objectives of Study.........................................................................................................4
1.5 Scope of Study................................................................................................................4
1.6 Justification of Study ......................................................................................................5
CHAPTER TWO: LITERATURE REVIEW............................................................................6
2.1 Electricity in Nigeria.......................................................................................................6
2.2 Self-Generation of Power ...............................................................................................9
2.2.1 Internal Combustion Engines ................................................................................10
2.2.2 Gas Turbines..........................................................................................................12
2.2.3 Solar-PV Cell.........................................................................................................13
2.2.4 Wind Turbine.........................................................................................................14
2.2.5 Fuel Cells...............................................................................................................15
2.3 Multicriteria Decision Making (MCDM) .....................................................................16
2.3.1 Introduction ...........................................................................................................16
2.3.2 Decision Making....................................................................................................17
2.3.3 Basic Concepts of Multicriteria Decision-Making................................................18
2.3.4 Terminologies........................................................................................................21
2.3.4.1 Alternatives............................................................................................................21
2.3.4.2 Criteria...................................................................................................................21
2.3.4.3 Attributes ...............................................................................................................21
2.3.4.4 Objectives ..............................................................................................................21
2.3.4.5 Decision Variables.................................................................................................22
2.3.4.6 Constraints.............................................................................................................22
2.3.4.7 Optimal solution ....................................................................................................22
2.3.4.8 Ideal solution .........................................................................................................22
2.3.4.9 Nondominated solution..........................................................................................22
2.3.5 Classes of MCDM .................................................................................................23
2.3.5.1 Multi Attribute Decision Making (MADM)..........................................................23
2.3.5.2 Multi Objective Decision Making (MODM).........................................................23

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2.3.6 Nondominance and Pareto Optimality ..................................................................24
2.3.7 Available MCDM techniques................................................................................25
2.4 Compromise Programming...........................................................................................25
2.5 The VIKOR Method .....................................................................................................26
2.6 Analytic Hierarchy Process (AHP)...............................................................................27
2.7 Fuzzy Logic ..................................................................................................................29
CHAPTER THREE: METHODOLOGY ................................................................................31
3.1 Background...................................................................................................................31
3.2 Theoretical Framework.................................................................................................32
3.3 Model Description ........................................................................................................32
3.3.1 Objective................................................................................................................32
3.3.2 Alternatives............................................................................................................33
3.3.3 Criteria...................................................................................................................33
3.3.4 Constraints.............................................................................................................33
3.4 Determination of Criteria Weights................................Error! Bookmark not defined.
3.4.1 Fuzzy Arithmetic .................................................Error! Bookmark not defined.4
3.4.2 Fuzzy Analytic Hierarchy Process (FAHP).........Error! Bookmark not defined.5
3.5 The VIKOR Method .....................................................................................................37
3.6 Summary of the Procedure............................................................................................39
CHAPTER FOUR: DATA COLLECTION, ANALYSIS AND APPLICATION..................41
4.1 Selection of Criteria ......................................................................................................41
4.1.1 Setup Cost..............................................................................................................41
4.1.2 Annual Maintenance and Operation Costs ............................................................42
4.1.3 Pollution from Exhaust Fumes ..............................................................................42
4.1.4 Noise......................................................................................................................42
4.1.5. Capacity Factor......................................................................................................43
4.2 Identification of Alternatives........................................................................................43
4.3 Determination of Criteria Weights................................................................................44
4.4 VIKOR Computations ..................................................................................................49
4.5 Results and Discussion .................................................................................................52
CHAPTER FIVE: CONCLUSION AND RECOMMENDATIONS ......................................55
5.1 Conclusion ....................................................................................................................55
5.2 Recommendations.........................................................................................................56

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REFERENCES ........................................................................................................................57
APPENDIX..............................................................................................................................62
I. Calculation of Costs and other Data in Table 4.1.............................................................62
II. Fuzzy Calculations of the Fuzzy Analytic Hierarchy Process .........................................65

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LIST OF TABLES
Table 2.1: MADM vs. MODM (Source: Hwang & Yoon, 1981)……………………………….…24
Table 3.1: Membership function of linguistic scale (Sun, 2011)…………………….….…………36
Table 4.1: Decision Maker 1's Pair-wise Ratings………………...…………………..……………….45
Table 4.2: Decision Maker 2's Pair-wise Ratings…...……………………………………..………….45
Table 4.3: Decision Maker 3's Pair-wise Ratings………………...……………………………..…….45
Table 4.4: Matrix M……………………………………………...………………………………………….47
Table 4.5: Fuzzy Geometric Means and Criteria Weights…………………………….………...…..48
Table 4.6: Best Non-Fuzzy Performance Values of the Criteria……………………………..…….48
Table 4.7: Values of the Alternatives with respect to each Criteria….............................................49
Table 4.8: Best and Worst Values of each Criterion Function………….………………..………...50
Tablee 4.9: Sj and Rj Values of the Alternatives……………..………….………………..……………51
Table 4.10: Qj values of the Alternatives (Rankings are in brackets)……….…...………….…….52
Table 4.11: Sj, Rj and Qj Rankings of the Alternatives………………………………………………...…...53
Table 4.12: Compromise Set of Alternatives…………………………………………………….………….53

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LIST OF FIGURES
Figure 2.1: Indicator of Electricity Crisis in Nigeria 1970 to 2004 (Iwayemi, 2008)……..…….7
Figure 2.2: Electricity Demand Projection in Nigeria (Sambo, 2008)………………………...……7
Figure 2.3: Gas Turbine Schematic...…………………………………..…..……………………………13
Figure 3.1: Flowchart of the VIKOR Method………………………………………………………….40

CHAPTER ONE
INTRODUCTION
1.1 Overview
Electricity plays a most important role in the socio-economic and technological development
of any country. It is more so essential for the survival of businesses, particularly production
and manufacturing systems that require large amounts of energy for their various activities.
Without doubt, energy, in the appropriate form and quantity, constitutes a major input to any
production system. Electricity is the most widely used form of power in the country. Its ready
availability at a reasonable price is a requirement for optimum productivity of any system.
1.2 Situation of Generation and Distribution of Electricity in Nigeria
It is a well known fact that Nigeria is faced with acute electricity problems, which are stalling
her development despite the vast human and material resources available in the country.
Adequate energy is an important input factor in any production process and an indispensable
factor in social and economic development and, consequently, the overall quality of life of
the population. It is widely accepted that there is a correlation between socio-economic
development and the availability of electricity (Sambo, 2008). Countries in the developed
world have installed capacities to generate more electricity than is needed for economic
progress. However, due to substantial investment in infrastructural development in the sector,
the demand for electricity in Nigeria far outstrips its supply and the little available supply is
epileptic in nature (Sambo, 2008). This is the crux of the problem.

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The situation is understatedly pathetic. A brief overview of the situation as is follows
(Osunsanya, 2008):
1. The Nigerian Electricity Supply Industry (ESI) is dominated by a state monopoly
known as the Power Holding Corporation of Nigeria (PHCN).
2. Only 36% of the populace is connected to the national grid.
3. Generation is currently between 2500MW – 3500MW compared to an installed
capacity of 5963MW. This is some improvement from the 1999 performance of
1300MW.
4. About 2500MW of self generation from petrol & diesel power generators exist.
5. Transmission lines are poorly maintained and frequently vandalized which results in
transmission losses of over 25% of electricity produced
6. Undersupply is underscored by the huge (40%) privately-owned alternative capacity
(diesel/petrol generators). This alternative capacity is supplied at a premium of up to
400% of grid price. Currently, industrial consumers (who can afford this) own the
bulk of the alternative capacity.
7. Demand has grown at a rate of 8.2% per annum since 1984 against GDP growth of
about 3-5% (Source: FGN/NNPC/EM Nigerian Gas Utilization Study).
According to the Electric Power Sector Reform Implementation Committee (2004), 2400MW
of electricity was being generated by generating sets in August 2000. These generating sets
vary in size, type and capacity. Some are small and light-duty (1.5-4.5 KVA) used in small
residential homes while others are large and heavy-duty (500-10,000 KVA). These generators
primarily use fossil fuels such as petrol, diesel and natural gas for power generation.

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1.3 Problem Statement
As far as most industrial outfits are concerned, continuous supply of electricity is an absolute
necessity. This is especially so in production setups which operate process lines and are
involved in the mass production of various goods. Sudden power outages may result in the
stoppage of product lines and hence huge costs are incurred in the form of damaged goods
and equipment, idle workers, re-setup costs, and so on. As a result, most industrial/production
outfits in the country are left to make a decision pertaining to self-generation of power.
The decision to privately generate power is one which requires careful study and analysis.
This is because, for most firms, the costs involved may significantly raise the initial start-up
costs of the business. Added to that is the annual maintenance and fuel costs. Incidentally,
indigenous, small-scale enterprises are worse affected. (Lee & Anas, 1991) reports that small-
scale enterprises may spend as much as 25% of the initial investment on self-provision of a
generator. Indeed, costs are a major factor in making decisions regarding self-generation of
power.
In recent times, however, with the emergence of globalization, requirement of stricter
adherence to industrial safety and health procedures and tighter environmental laws, more
factors/decision criteria need to be considered in making a decision on the method of self-
generation of power especially on a large scale. This does not mean that cost seizes to be a
main factor. In fact, it still remains a determining one. However, there are other important
considerations such as environmental and safety factors in the workplace. This situation is
even more common in multi-national companies that are responsibly dedicated to
international agreements to reduce emissions of greenhouse gases. As a result, the situation is
one of Multi-Criterion Decision Making (MCDM).

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In MCDM, the question is not to obtain the best solution that satisfies all decision
criteria/factors as is the case in single objective optimisation. This is because the optimisation
of a particular criterion is usually done to the detriment of one or more other criteria. In
reality, human decision-making behaviour hardly ever considers only one aim or objective at
a time. Instead, the human mind always seeks for the best way, method or solution that
satisfies one or more criteria. MCDM is not just about maximizing or minimizing a single
goal but searching for stable patterns of harmony among all goals because some goals are in
conflict with others (Zeleny, 1974). In this particular situation, we aim to select the method of
self-generation of electricity that best „optimises‟ all the various decision criteria which shall
be identified.
1.4 Objectives of Study
The primary objectives of this work are as follows:
1. Identify the various feasible methods of electricity generation for an industrial facility
2. Identify the relevant decision criteria and their weights
3. Determine the electricity generation method(s) which serve as the „best compromise‟
that satisfies the considered criteria.
1.5 Scope of Study
This project gives a quick overview of the energy situation in the country as a background to
the necessity of self-generation of electricity by commercial and industrial setups. This is
done by reviewing the available literature and statistics about the subject.
The study is based on the requirements of a large-scale manufacturing plant in Ibadan. As a
result, the energy requirements are much more than that of a typical small or medium-sized
facility. Another consequence is that the study is limited to the feasible methods of power
generation in the city given its location and available resources. In this study, the VIKOR

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(Vlse Kriterijumska Optimizacija Kompromisno Resenje which means multicriteria
optimization and compromise solution) method is the multi-criteria decision making tool of
choice. It is a compromise ranking method based on the much wider Compromise
Programming (CP) algorithm. Other applicable methods were briefly mentioned. However,
this study aims to demonstrate the use of the VIKOR method. Consequently, a small number
of criteria (about 5) are used in making the decision. The methodology is, nonetheless,
explained in a clear and understandable way.
1.6 Justification of Study
The issue of self-generation of electricity is prevalent in Nigeria. There is hardly a
production/manufacturing setup that does not encounter this decision at least once. As such, a
critical study of decisions involving the method of self-generation of electricity in the
presence of a number of decision criteria is, to say the least, necessary. The importance of
energy to the productivity and continuous sustenance of a production facility requires that
this decision be as accurate as it can possibly be.
The VIKOR method has been chosen from the various decision making techniques available.
This is due to its efficiency in solving discrete decision problems with noncommensurable
and conflicting criteria (Opricovic and Tzeng, 2004). Some fuzzy logic was incorporated in
the determination of criteria weights so as to factor in the subjectivity of the human rationale.

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CHAPTER TWO
LITERATURE REVIEW
2.1 Electricity in Nigeria
The history of electricity in Nigeria dates back to 1896 when electricity was first produced in
Lagos, barely fifteen years after its introduction in England (Sambo, 2008). Despite the fact
that its existence in the country is over a century, its development has been at a rather slow
rate. For over twenty years prior to 1999, the power sector did not witness any substantial
investment in infrastructural development. During that period, new plants were not
constructed and the existing ones were not properly maintained, bringing the power sector to
a most deplorable state. In 2001, generation went down from the installed capacity of about
5,600MW to an average of about 1,750MW, as compared to a load demand of 6,000MW. In
addition, only nineteen out of the seventy-nine installed generating units were in operation.
The overall consequence is the unreliability of the supply from the public grid. As such, there
is the need to make provision for alternative(s).
The situation at present is underscored by the huge (about 40%) privately-owned alternative
capacity (mainly diesel/petrol generators). This alternative capacity is supplied at up to 400%
of grid price (Osunsanya, 2008). The bulk of this generation is by large commercial and
industrial outfits.
Much has been already been said and written about the state of power in Nigeria. The impact
on social and economic concerns in the country has also been extensively discussed in
various literatures. While some have tried to expose the structural complexities that seem to
be preventing any real development in the sector, others have focused on proffering the
way(s) out of the current debacle. There has also been a vast amount of work done on the
economics of shortage of power supply. These and more shall be briefly reviewed in this

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section. Figure 2.1 shows the situation in the country with regards to capacity utilization and
related losses. The projected demand for specified future periods are shown in Figure 2.2.
Figure 2.1: Indicator of Electricity Crisis in Nigeria 1970 to 2004 (Iwayemi, 2008)
Figure 2.2: Electricity Demand Projection in Nigeria (Sambo, 2008)

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Despite the fact that Nigeria has vast reserves of crude oil and natural gas, coal, tar sands and
renewable energy resources such as hydro, fuel wood, solar, wind and biomass (National
Energy Policy, 2003), she still remains largely incapable of generating an adequate amount of
energy to meet her local demands. Most of the available supply comes from hydro sources,
crude oil and natural gas. Two other sources: wood fuel and solar are used in their crude
forms for heating, cooking and lighting (Ayodele, 2003). The most obvious indicator of the
situation is the massive load shedding. These power outages, of course, come with their costs.
For the industrial sector, existing measure of outage costs vary between $1.27 to $22.46/kWh
of unserved electricity. Residential outage costs vary between $0.02 and $14.61/kWh
unserved (Caves et al, 1992).
There are essentially five ways by which firms may tackle unreliable electricity supply. These
are choice of location, factor substitution, private provision, choice of business and output
reduction (Adenikinju, 2005). While all these elements are presently observed among
Nigerian firms, the most common approach has been through private provision. Electricity
consumers have responded to the PHCN‟s inefficiency through self-generation. Electricity
users, both firms and households, now find it necessary to provide their own electricity in
part or in whole to substitute or complement PHCN supply by factoring generator costs into
the overall investment cost, thus raising significantly the set-up cost for manufacturing firms
operating in the country (Adenikinju, 2005). A few studies have tried to measure the cost of
electric power shortages in Nigeria. Adenikinju (2005) cited such works as (Ukpong, 1973),
(Iyanda, 1982) and (Lee & Anas, 1992). It has been stated that the poor state of infrastructure
supply in developing countries has a negative impact on their economic performance. For
instance, Lee and Anas (1992) report that manufacturing establishments in Nigeria spend on
average 9% of their variable costs on infrastructure, with electric power accounting for half
of this share. This, however, is not the focus of this study.

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2.2 Self-Generation of Power
Self or own-generation of power simply refers to the in-house generation of electricity by
individuals or commercial bodies for their own activities as against acquiring power from the
public grid. This is usually done on a much smaller scale than that of the public grid and at a
significantly higher cost (Foster & Steinbuks, 2009). It has also been found that self-
generation of electrical power accounts for only around 6 percent of the installed generating
capacity in Sub-Saharan Africa (Foster & Steinbuks, 2009). Lee and Anas (1991) identified
four different private response strategies pursued by firms:
 Self-sufficiency: In this case, the firm provides its own infrastructural services to the
point where it does not need any public input. This is common in industries or setups
that incur huge losses due to shortage of power supply.
 Stand-by private provision: Here, the firm has its own infrastructural facilities in place
and switches to these facilities where the quality or reliability of the public service
falls below a critical level. This is a strategy adopted by a lot of industrial/production
setups in Nigeria.
 Public source as standby: The firm relies primarily on its own facilities but switches
to the public supply during those times of the day when the public source delivers a
high quality service.
 Captivity: The firm continues to rely on the public source exclusively despite the very
low reliability of such services.
The unreliability of supply from the public grid has led most manufacturers to incur extra
costs for private alternatives. As a result, the generator market is very vibrant. Adenikinju
(2005) did some commendable work about the statistics of self-generation of power in

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industrial facilities in Nigeria. His findings show that there is a large percentage of self-
generation as only 6.2% rely exclusively on the public grid.
These private generators are of various kinds and utilize different kinds of fuels which
include diesel, petrol and natural gas. The cost of these fuels and other operating and
maintenance costs also add to the huge expenses associated with self-generation of power. By
taking advantage of the huge economies of scale in the industry, PHCN is able to supply
electricity at a much lower cost than private provision. This cost differential is large,
sometimes running to over four times. A 1983 joint UNDP/World Bank study estimated a
cost differential of 16–30% for large industrial establishments in the country with auto-
generation. In spite of this large cost differential, however, over 90% of Nigerian
manufacturers make provision for auto-generation (Adenikinju, 2005).
In addition, the various occupational health and safety risks associated with these electricity
generation facilities are quite substantial. Noise, pollution fumes and the safety of storing and
handling fuel are important issues to consider while planning for a private generating system.
The feasible means of self-generation of electricity in Nigeria are discussed as follows.
2.2.1 Internal Combustion Engines
These are also called reciprocating engines. They are, by far, the most common and most
technically mature of all the available electricity generation technologies. They are available
as in sizes/capacities ranging from light-duty engines (e.g. 2-5 kW for residential back-up
generation) to large industrial generators (0.25-7 MW).
Reciprocating engines use commonly available fuels such as gasoline (petrol), natural gas,
and diesel fuel. A reciprocating, or internal combustion, engine converts the energy contained
in a fuel into mechanical power which is then used to turn a shaft in the engine. A generator
attached to the internal combustion engine converts the rotational motion of the shaft into
power.

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There are two methods for igniting the fuel in a reciprocating engine: spark ignition and
compression ignition engines. In a spark ignition engine, a spark is introduced into the
cylinder by a spark plug at the end of the compression stroke. This ignites the fuel-air mixture
which results in its expansion and then pushes a piston to do some work. Fast-burning fuels,
like gasoline and natural gas, are commonly used in such engines. In compression ignition
engines, the fuel-air mixture spontaneously ignites when the compression significantly raises
its temperature. These engines work best with slow-burning fuels, like diesel. While gas
generators are frequently used in small units and are environmentally friendly, diesel generators are
preferred for large applications (above 150 kW) due to their significant cost advantage. This is
because gas is relatively cheaper than diesel fuel. Moreover, gas generators usually have an extended
runtime as a result of the endless supply of natural gas from pipelines.
There are also some reciprocating engines which have either been designed or modified to
run on two fuels, the most common being diesel and natural gas. These engines are known as
dual-fuel or bi-fuel engines. Bi-fuel generators usually are conventional, high-volume diesel
engines that are modified for bi-fuel operation. The diesel fuel enters the engine through the
injection system. As with a standard diesel engine, there are no spark plugs. The ignition of
the diesel fuel provides the spark required by the natural gas. The natural gas is later
introduced when the compression has suitably increased the temperature. This type of
engines was designed as a kind of compromise in order to utilize the best of both diesel and
gas engines. This is so because both have their merits and shortcomings.
Since reciprocating internal combustion engines are the most common and developed
technology for private power generation all over the world, they have the lowest initial setup
costs. The capital cost of a gas generator is about twice that of a diesel engine of the same
capacity. However, natural gas is usually less expensive than diesel fuel for the same heat
content. Hence, fuel costs in gas engines are significantly lesser than in diesel engines.

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Overall engine cost ($/kW) increases with size. Despite their availability, reciprocating
engines are major sources of emission of greenhouse gases, especially CO2 and NOx gases.
This is a major concern in most advanced economies.
2.2.2 Gas Turbines
A turbine, like an internal combustion engine, is essentially a large compressor. However, the
turbine is a continuous flow/fuel-burning machine whereas an internal combustion engine
relies on serial piston strokes to maintain air and fuel flow. Gas turbines are currently the
most common prime mover in larger-scale power generation, especially where natural gas is
available at significantly lower costs than those of solid fuels. Gas turbines are available in
sizes ranging from 500 kilowatts (kW) to 250 megawatts (MW). Gas turbines can be used in
power-only generation or in combined heat and power (CHP) systems.
For operation, intake air passes through a compressor before being heated by the combustion
of the fuel. The expanding air is then used to drive a turbine before exiting through the
exhaust and heat processes (see figure 2.3). Compressors require a large amount of energy,
making the choice of compressor crucial to the overall efficiency of the turbine. Natural gas
is the main fuel source, but other fuels can be used. Gas turbines are a mature and
economically efficient technology with broad acceptance in the electricity market place.
There is a class of gas turbines known as microturbines which are essentially very small gas
turbines with outputs of about 30 kW to 250 kW. These types of turbines evolved from
automotive and truck turbochargers on board aircraft and small jet engines. Due to their very
low emissions and low maintenance requirements, microturbines are well suited for small-
scale power generation. Their costs, however, and competing piston and diesel engines in the
same power class or higher, have long made them uneconomical. As their prices per kilowatt
drop, they will find greater acceptance.

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Figure 2.3: Gas Turbine Schematic
Gas turbines remain one of the cleanest means of generating electricity, with emissions of
oxides of nitrogen (NOx) from some large turbines in the single-digit parts per million (ppm)
range. Because of their relatively high efficiency and reliance on natural gas as the primary
fuel, gas turbines emit substantially less CO2 per kilowatt-hour (kWh) generated than any
other fossil technology in general commercial use.
2.2.3 Solar-PV Cell
A solar-PV system is one which uses one or more photovoltaic cells (solar panels) to convert
sunlight directly into electricity. A single photovoltaic cell will typically produce about 1 to 2
DC watts. In order to increase the power output, several cells are interconnected to form a
module. Similarly, modules can be connected to form an array. Several arrays covering
thousands of square metres are usually needed to produce power on a large scale. The PV
system consists of multiple components such as the photovoltaic modules, mechanical and
electrical connections and a means of regulating the output.
The output and performance of PV systems depend on a number of factors, most prominent
of which is the amount of available sunlight. Shades, dirt and cloud cover significantly reduce
their output. Commercially available photovoltaic modules range from about 5 to 15%

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efficiency at converting sunlight into energy. Efforts are currently under way to improve
photovoltaic cell efficiencies.
PV systems are best in regions around the equator where there is a longer period of sunlight.
Nigeria lies within a high sunshine belt and thus has enormous solar energy potentials.
Average sunshine hours are estimated at 6 hours per day (Sambo, 2009). Due to the non-
availability of continuous sunlight throughout the day, PV systems are usually operated as
hybrid systems. A hybrid system combines PV with other forms of generation, usually a
diesel generator.
Solar-PV system costs have substantially reduced over the past 20 years. However, setup
costs still remain quite high. However, recent trends towards environmental sustainability
will make this technology a very important one in the not too distant future.
2.2.4 Wind Turbine
Wind energy is a source of renewable power which comes from air current flowing across the
earth's surface. Wind has been utilized by man since early times. Sail boats and wind mills
are examples of how wind energy has been applied in the past. Wind turbines for electricity
generation are a more recent approach to wind utilization. Wind turbines extract the energy
from the wind by transferring the momentum of the air passing through the wind turbine
rotor, into the rotor blades. The rotor blades are aerofoil, and used for concentrating the
energy of the moving air into a single rotating shaft. The mechanical power of the shaft can
then be harnessed by coupling it with an alternator for electricity generation. Wind energy is
clean, free and inexhaustible.
The turbine tower height is an important factor which has to be considered while selecting the
type of the turbine. The reason for this is that there is a considerable change in the wind
velocity profile at different heights. The higher the turbine tower height, the higher the wind
speeds due to reduced obstruction by other buildings, trees etc; and the higher the power

15
generated by the turbine. In Nigeria, wind is available at annual average speeds of about 2.0
m/s at the coastal region and 4.0 m/s at the far northern region of the country. Assuming an
air density of 1.1 kg/m3
, wind energy intensity, perpendicular to the wind direction, ranges
between 4.4 W/ m2
at the coastal areas and 35.2 W/ m2
at the far northern region (Sambo,
2009).
Investment in wind turbines is a very capital-intensive venture, usually running into over a
million dollars per MW. The limited life span of wind turbine when compared with hydro and
thermal stations makes such investments by private bodies difficult and rare. A modern wind
turbine is designed to work for an average of 120 000 hours throughout its estimated life-span
of about 20 years. This would be the turbine operating for approximately 66% of the time for
two decades. From experience, the maintenance costs of a new turbine will be very low but as
the turbine ages these costs will increase. The estimated maintenance costs of modern
turbines are in the range of 1.5% to 2% of the original investment per annum (WMI, 2010).
Wind energy has the added advantage of no harmful emissions and requirement for fuel.
2.2.5 Fuel Cells
These are electrochemical cells that convert the chemical energy in a fuel into electric energy.
Much like conventional electrochemical cell batteries, they are, however, different because
they receive reactants from external sources, which must be replenished. In essence, fuel cells
are electrochemical devices in which fuel is combined with oxygen from the ambient air to
produce electricity and heat. The non-combustive process is a form of direct fuel-to-energy
conversion and is much more efficient than traditional fossil fuel power plants. As a result,
CO2 emission is reduced and the absence of combustion prevents the production of NOx and
particulate matter such as soot. The most common fuels used in commercial fuel cells are
hydrogen and natural gas.

16
Fuel cells incorporate an anode and a cathode, with an electrolyte in between, similar to a
battery. The material used for the electrolyte and the design of the supporting structure
determine the type and performance of the fuel cell. Several types of fuel cells are currently
being developed. These include Proton-Exchange Membrane (PEM) fuel cells, Solid-Oxide
Fuel Cells (SOFCs), Alkaline Fuel cells (AFCs), Phosphoric Acid Fuel Cells (PAFCs),
Molten-Carbonate Fuel Cells (MCFCs), and Direct-Methanol Fuel Cells (DMFCs) (Lipman
et al, 2004). Commercial fuel cell power generation plants consist of several of these fuel
cells arranged in stacks to provide the required system voltage and power. They also
comprise the equipment required to provide the proper gas flow and power conversion cells.
With availability ratings better than 90%, fuel cells are not affected by such external
influences which affect other environmentally-friendly technologies like wind turbines and
solar arrays. However, the main constraint to their market-wide acceptance remains the huge
capital costs involved (Smit, 2002).
2.3 Multicriteria Decision Making (MCDM)
2.3.1 Introduction
During the past decades, Operations Research (OR) has come a long way as a field that
enhances scientific management of people and processes. Within the OR field, various
interconnected areas of study have been developed on the basis of different decision-making
methods and contexts. OR is mainly involved with model building and algorithmic
optimization procedures that facilitate the analysis of complex real-world problems.
An important and rather common feature implication of real-world decision-making problems
is their multidimensional character, which often requires the consideration of multiple
conflicting points of view, even in situations where a single decision maker is involved.
Nowadays, economic, social, and environmental criteria are taken into consideration in
practically all decision situations, in order to adequately describe the diverse outcomes of the

17
available alternatives. Within this context, an effective decision process should naturally
explore the conflicting nature of the criteria, the corresponding tradeoffs, the goals set by the
decision makers, and of course the way that these can be introduced in an appropriate
decision model that takes into account the bias of the decision process and the decision
maker(s). Nevertheless, with the introduction of multiple points of view, criteria, and factors,
universally acceptable (objective) solutions are no longer feasible. While this may be
cumbersome, it highlights the difficulty of decision-making in a realistic context.
2.3.2 Decision Making
A decision making situation occurs when there exist an objective to be reached in the
presence of alternative courses of action and a variety of factors that are relevant to the
evaluation of the alternatives or their probability of success. Decision making is the study of
identifying and choosing alternatives based on the values and preferences of the decision
maker(s). Making a decision implies that there are alternative choices to be considered, and in
such a case we want not only to identify as many of these alternatives as possible but to
choose the one that best fits with our goals, objectives, desires, values, and so on (Harris,
1980).
Decision making can be shortly defined as the cognitive process based on explicit
assumptions, which leads to the selection among feasible alternatives up to a final choice
(Trincas, 2010). Structured rational decision making is an important part of all science-based
activities, where specialists apply their knowledge in a given area to make decisions. MCDM
tools are also employed by management in making strategic decisions.
A strategic decision has been defined as one that is “important, in terms of the actions taken,
the resources committed, or the precedents set” (Mintzberg et al, 1976). Strategic decisions
are “infrequent decisions made by the top leaders of an organisation that critically affect
organizational health and survival” (Eisenhardt & Zbaracki, 1992). Furthermore, the process

18
of creating, evaluating and implementing strategic decisions is typically characterised by the
consideration of high levels of uncertainty, potential synergies between different options,
long term consequences, and the need of key stakeholders to engage in significant
psychological and social negotiation about the strategic decision under consideration. The
fact that strategic decisions typically involve the consideration of multiple strategic objectives
suggests the adoption of MCDM as the evaluation tool for strategic choices.
Usually, the alternative that best satisfies one individual requirement does not have the best
performance on other requirements. That is, typically there is no design that has the best
performance on all the requirements. As a result, compromises need to be made when
multiple criteria are simultaneously taken into account.
2.3.3 Basic Concepts of Multicriteria Decision-Making
Multicriteria decision-making methods apply to problems where a decision maker (DM) is
selecting or ranking a finite number of alternatives which are measured by often conflicting
attributes. Multiple criteria pervade all that people do and include such public policy tasks as
determining a country‟s policy, developing a national energy plan, as well as planning
national defence expenditures, in addition to such public/private company tasks as are
product development, pricing decisions, and research project selection. All have a common
theme, i.e. multiple conflicting targets. It is often the case that good values of some criteria
inevitably go with poor values of others in the same alternative option, so that the best option
is always a compromise in some sense. In order to find the best compromise solution,
decision makers are required to take all the criteria into account concurrently when making
decisions.
Typically, in order to solve an MCDM problem, Trincas (2010) identified some necessary
factors which need to be known beforehand:
 well defined, measurable criteria;

19
 preference information on the criteria
 feasible design alternatives
 rational decision–making method.
The criteria can be thought of as the measure of performance for an alternative, such as
corrosion resistance and thermal expansivity for an oil pipeline, and can be checked with
respect to the DM`s requirements.
The alternatives are the candidates among which the „best solution‟ is selected. They may be
designs that already exist, or need to be generated in the design process. Since the criteria do
not have the same priority to the DM, the preference information on the criteria should be
defined. Relative weights, which are assigned beforehand or calculated, are an objective way
of representing preference information. A set of appropriate alternatives has critical impact
on the final solution because the final solution is one of the elements of this set.
MCDM usually refers to the set of methods enabling a decision maker to make decisions in
the presence of multiple, often conflicting, criteria. It is an excellent tool for multiattribute
selection and multiobjective optimization of industrial products. MCDM as a discipline, and
its application, has increased significantly after development of computer science, as most of
methods are complex combinations of higher mathematics.
According to Trincas (2010), MCDM techniques share the following common characteristics:
 Problem statement: this is based on identifying the true needs of the DM and
formulating them in a set of targets (attributes, objectives) for the compromise
solution. The problem statement has to specifically express what is needed to achieve
the established goals. A good problem statement plays an important role in
determining the success of the final solution.
 Resolution of conflict among multiple criteria: The problem definition yields a set of
criteria on which the DM should base his decisions. Criteria play the essential role in

20
the decision–making process, where an alternative solution is deemed successful if the
customer desired levels are met. Multiple criteria usually conflict with each other.
MCDM allows managing these conflicts since it is a conflict–resolution approach.
 Normalization of attribute values: Each attribute/objective has a different unit of
measurement. In a technical system selection case, fuel consumption is expressed by
tons per mile, comfort is measured by specialized indexes in a non-numerical way,
cost is indicated by monetary units, etc. Hence, a normalization of the criteria values
may be essential to obtain comparable scales.
 Selection/Optimization: Solutions to design problems are either to select the best
solution among previously defined finite number of alternatives or to optimize the
„best possible solution‟. At first, the MCDM selection process involves searching for
an alternative that is the „best possible solution‟ or the „preferred solution‟ over all
criteria. Then the „preferred solution‟ can be improved by means of a MCDM
optimization process.
In summary, a multicriteria decision-making process comprises a decision maker (DM), or a
group of DMs who make the decisions, a set of objectives that are to be pursued and a set of
alternatives from which one is to be selected. In a decision situation the DMs have to manage
goals, criteria, objectives, attributes, constraints and targets, in addition to decision variables.
Although goals, criteria, objectives, and targets have essentially similar dictionary meanings,
it is useful to distinguish them in a decision-making context. For example, while criteria
typically describe the standards of judgements or rules to evaluate feasibility, in MCDM they
simply indicate attributes and objectives.

21
2.3.4 Terminologies
There are a number of terms used in MCDM literature. These include alternatives, criteria,
attributes, objectives and so on. There are no universal definitions of these terms, since some
authors make distinctions in their usage, while others may use them interchangeably.
2.3.4.1 Alternatives
Alternatives are the finite set of different design solutions which are available to the decision
maker. They are simply the options available to the DM which are capable of solving the
problem at hand, at least to some extent.
2.3.4.2 Criteria
Criteria are a measure of effectiveness of performance. They are the basis by which the
performance of an alternative is evaluated. Criteria may be in the form of attributes or
objectives.
2.3.4.3 Attributes
Attributes are generally referred as “designed-to criteria” that describe the performance,
properties, and alike, of a technical system (size, weight, range, speed, payload, reliability,
cost, etc.). They provide a means of evaluating the levels of aspiration achieved on various
targets. That is why they are often referred as soft constraints. Each design alternative can be
characterized by a number of attributes chosen by the DM.
2.3.4.4 Objectives
Objectives are unbounded, directionally specified (maximization/minimization) requirements
which are to be pursued to the greatest extent possible. It is very likely that objectives will
conflict with each other in that the improved achievement with one objective can only be
accomplished at the expense of another. They generally indicate the desired direction of
change, i.e. the direction in which to strive to do better as perceived by the decision maker.

22
2.3.4.5 Decision Variables
A decision variable is one of the specific choices made by a decision maker. For example, the
weight of an industrial product is a decision variable.
2.3.4.6 Constraints
Constraints are temporarily fixed requirements on attributes and decision variables which
cannot be violated in a given problem formulation, that is, upper and lower bounds cannot be
exceeded, and strictly requirements must be satisfied precisely. Constraints divide all possible
solutions (combinations of variables) into two groups: feasible and infeasible. They are crude
yes or no requirements, which can be either satisfied or not satisfied.
2.3.4.7 Optimal solution
An optimal solution to a MCDM problem is one which results in the maximum value of each
of the attribute or objective functions simultaneously. That is, x*
is an optimal solution to the
problem if and only if x*
ϵ X and f (x*
) ≥ f (x) for all x*
ϵ X.
Since it is the nature of MCDM criteria to conflict to each other, usually there is no optimal
solution to a MCDM problem.
2.3.4.8 Ideal solution
The concept of the ideal solution is essential for the approach of multicriteria decision
making. An ideal solution may be indicated also as optimal solution, superior solution, or
utopia. Though an ideal solution does not actually exist, the concept of an ideal solution is
essential in the development of MCDM methods. For example, a compromise model is based
on the idea of obtaining the „best possible solution‟, which is the closest to the ideal solution.
2.3.4.9 Nondominated solution
This is called various names in different disciplines like non-inferior solution and efficient
solution in MCDM, a set of admissible alternatives in statistical decision theory, and Pareto–
optimal solution in economics.

23
A feasible solution x*
in MCDM is called a nondominated solution if and only if there exists
no other feasible solution that will yield an improvement in one attribute without causing a
degradation in at least another attribute (Trincas, 2010). In other words, a nondominated
solution is achieved when no attribute can be improved without simultaneous detriment to at
least another attribute.
2.3.5 Classes of MCDM
MCDM techniques fall into two broad classes: the Multi Attribute Decision Making
(MADM) and Multi Objective Decision Making (MODM) techniques (Pohekar &
Ramachandran, 2004).
2.3.5.1 Multi Attribute Decision Making (MADM)
This includes methods that involve selection of the „best possible design‟ from a discrete pool
of alternatives described in terms of their prioritized attributes. Attributes are generally
defined as characteristics that describe in part the state of a product or system. Assessment of
alternatives and selection of the „best possible design‟ is done via straight-forward evaluation.
The multiple attribute techniques either directly ask the DM for an assessment of the
strengths of these preferences or they infer them from his/her past choices, while all attributes
are evaluated simultaneously.
2.3.5.2 Multi Objective Decision Making (MODM)
This relates to techniques that synthesize a set of alternatives, which optimize or „best satisfy‟
the set of mathematically prescribed objectives (or goals) and constraint functions of the DM.
MODM problems involve the design of the „best alternative‟ by considering the trade-offs
within a set of interactive design constraints. They assume continuous solution spaces i.e. the
number of alternatives is effectively infinite and the trade-offs among design objectives are
typically described by continuous functions. Multicriteria optimization problems fall under
the heading of MODM (Stadler, 1988). That is, optimization will be performed to maximize

24
or minimize the associated objective(s), and the final selected solution is one with the best
values of the objective(s). Each optimization problem can be classified into two parts: the set
of functions to be optimized (minimized or maximized), i.e., objectives; and the set of
functions to be satisfied in terms of their predetermined values, i.e., constraints.
The main distinctions between MADM and MODM are enumerated in Table 2.1.
Table 2.1: MADM vs. MODM (Source: Hwang & Yoon, 1981)
Elements MADM MODM
Criteria Attributes Objectives
Objectives Implicit Explicit
Attributes Explicit Implicit
Alternatives Finite number Infinite number
Application Design Selection Design Optimization
2.3.6 Nondominance and Pareto Optimality
Since good values of some criteria inevitably go with poor values of others, the goal of the
MCDM is to find the „best compromise‟ solution which has best overall performance of
satisfying all the attributes. This „best compromise‟ solution can be obtained from a set of
design alternatives referred to as the efficiency frontier or Pareto optimal-set. All these
solution sets consist of points having a simple and highly desirable property, i.e. dominance.
A point in a set is nondominated in that no other point is feasible at which the same or better
performance could be achieved with respect to all criteria, with at least one being strictly
better.
The definition of the Pareto optimality indicates that there is no other feasible solution in the
design space which has the same or better performance than the Pareto optimal solution
considering all criteria; the Pareto–optimal solution does not have the best performance in all
criteria (Zeleny, 1982). It is clear that the Pareto-optimal solution is a nondominated solution

25
which is achieved when no criteria can be improved without simultaneous detriment to at
least one other criterion. The locus of the Pareto–optimal solutions is known as Pareto
frontier.
2.3.7 Available MCDM techniques
These are many and varied. (Adeyeye & Oyawale, 2010) identified some methods which
include Weighted Sum Scalarization (WSS) techniques, goal programming and compromise
programming. Others include nonlinear aggregation (desirability indices), Analytic
Hierarchical Process (AHP), Multi-Attribute Utility Theory (MAUT), ELECTRE I-III,
PROMETHEE, and cooperative game theory (Opricovic & Tzeng, 2004). However, this
work shall focus on the Compromise Programming (CP) technique and more specifically, the
Compromise Ranking method also known as the VIKOR method.
2.4 Compromise Programming
Compromise Programming (CP) was first proposed by Zeleny (1973). CP employs the
concept of distance to analyze multiple objective problems. This distance is not limited to the
geometric sense of distance between two points; it is rather used as a proxy to measure
degrees of human preferences. CP selects a nondominated preferred solution from a feasible
set, on the basis of the solution‟s closeness to an infeasible ideal point (Zeleny 1973). A
nondominated solution in a Multi Objective Decision Making (MODM) problem is one that
cannot produce any improvement in any one of the objectives without making at least one
other objective worse (Tecle et al, 1988), while an ideal point represents the joint location of
the individual maximum values of all the objectives. Therefore, arriving at a compromise
solution can be viewed as minimizing a Decision Maker (DM)‟s regret for not obtaining the
ideal solution.
Compromise programming involves two types of parameters. The first is the parameter p (1 ≤
p ≤ ∞) that reflects the importance of the maximal deviation from the ideal value. The second

26
is the weight wi, reflecting the relative importance of the i-th criterion to the decision maker.
Freimer and Yu (1976); Yu and Leitmann (1976) and Duckstein and Opricovic (1980)
indicated that the parameter p had a balancing effect on the utility and distance from the ideal
so that increasing p reduced utility but, at the same time, reduced the distance from the ideal
point.
Compromise programming can be used in both mathematical programming (design problem)
and decision analysis (Nachtnebel, 1994). Some previous works involving CP include a
decision problem of urban water management by Abrishamchi et al (2005), selection of an
industrial robot for a specific engineering application (Athawale et al, 2010), a study
evaluating water use in agriculture by Ganoulis (2001) and a research work into creating a
macroeconomic policy in a general equilibrium framework by Gagné et al (2005).
2.5 The VIKOR Method
The VIKOR (Vlse Kriterijumska Optimizacija Kompromisno Resenje, Serbian for
multicriteria optimization and compromise solution) method was proposed by Opricovic
(1998). The VIKOR method, a tool applicable in multi-criteria analysis, can identify a
compromise solution from amongst several alternatives in the presence of multiple criteria.
All alternatives are evaluated with regard to the identified criteria which carry equal or
varying weights. The compromise ranking is performed by comparing the measure of
closeness to the ideal alternative (Opricovic and Tzeng, 2004). This method can be employed
to solve MCDM problems with conflicting and non-commensurable (with varying units)
criteria, assuming that compromise can be acceptable for conflict resolution, when the
decision maker wants a solution that is the closest to the ideal solution and the alternatives
can be evaluated with respect to all the established attributes. It focuses on ranking and
selecting the best alternative from a finite set of alternatives with conflicting criteria, and on

27
proposing the compromise solution (one or more). Adeyeye and Oyawale (2010) stated the
conventional form of the parameter distance from the ideal, Lp,j as
1 ≤ p ≤ ∞; j = 1, 2,…, m
The compromise solution is a feasible solution, which is the closest to the ideal solution, and
a compromise means an agreement established by mutual concessions made between the
alternatives. The multicriteria merit for compromise ranking is developed from the Lp-metric
used in the compromise programming method (Zeleny, 1982). Yu (1973) introduced
compromise solutions, based on the idea of finding a feasible solution that is as close as
possible to an ideal point. Zeleny (1982) stated that alternatives that are closer to the ideal are
preferred to those that are farther away. To be as close as possible to a perceived ideal is the
rationale of human choice. As an aggregating function Yu (1973) introduced Lp metric for a
distance function, called the group regret for a decision, a regret that the ideal cannot be
chosen. Yu (1973) and Freimer and Yu (1976) indicated several properties of compromise
solutions, and the role of parameter p.
Some previous work that have effectively applied the VIKOR method are as follows: Cetin
and Cetin (2010) performed a financial evaluation of banks, selection of materials under
aggressive environments by Cristóbal et al (2009), prioritizing land-use restraint strategies by
Chang and Hsu (2009) and a hybrid performance evaluation system for notebook computer
companies in Sun (2011). Athawale and Chakraborty (2011) state that the main focus must
not lie on the selection of the most appropriate MCDM method adopted but on proper
structuring of the decision problem considering the relevant criteria and decision alternatives.

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2.6 Analytic Hierarchy Process (AHP)
The Analytic Hierarchy Process (AHP) method, originally developed by Saaty (1980), deals
with the study of how to derive ratio scale priorities or weights through pairwise relative
comparisons. Saaty and Sodenkamp (2010) stated the following about AHP:
“It is a psychophysical theory of measurement. This means that it makes the assumption that
judgments about subjective preferences and understanding are essentially not so different
from and depend on judgments about the physical world in which we acquire our experiences
and understanding. In the physical world, we respond to intensities of events, such as the
varying intensities of sight, sound and smell. These intensities fall in different threshold
intervals of just noticeable differences because we are unable to detect change in intensity
until a stimulus is increased by a noticeable amount. Judgments must reflect not only
knowledge about influences, but also the strengths with which these influences occur. These
strengths are expressed by us, and especially by experts who have experienced the
complexity with which we are concerned, through judgments from which priorities are
derived in relative form that reflect numerical intensities that can be validated in those cases
where we have measurement in order to improve our confidence in the applicability of our
quantified judgments in those cases where measurements are not available.”
The basic idea of AHP is the conversion of subjective assessments of relative importance to a
set of overall scores or weights. AHP not only supports the decision makers by enabling them
to structure complexity and exercise judgement, but allows them to make both subjective
preferences and objective evaluation measures in the decision process. The Analytic
Hierarchy Process (AHP) provides a comprehensive structure and mathematics to incorporate
measurements for tangible criteria and derives priorities for intangible criteria to enable one
to choose a best alternative for a decision (Saaty & Sodenkamp, 2010). It provides a useful
mechanism for checking the consistency of the evaluation measures and alternatives

29
generated by the design team thus reducing bias in decision making. AHP has been
extensively applied by academics and professionals, mainly in engineering applications
involving financial decisions associated to non-financial attributes (Malakooti, 1991).
In this work, the AHP is used in determining the weights of each criterion according to expert
advice. AHP is a very useful decision analysis tool in dealing with multiple criteria decision
problem, and has successfully been applied to many decision areas. However, perceived
inadequacies of the AHP especially due to inherent uncertainty and imprecision of the human
subjective decision making process have led to many a complaint. Some of these complaints
have pointed out that the AHP is mainly used in nearly crisp-information (data with absolute
certainty) decision applications; that it creates and deals with an unbalanced scale of
judgment; that it does not consider the uncertainty associated with the mapping of human
judgment to a number by natural language; that the ranking of the method is rather imprecise;
and that the subjective judgment by perception, evaluation and selection based on preference
of decision-makers have a great influence on the integrity of the AHP results (Ravi et al,
2008; Karsak, 2002).
In order to address these concerns as well as to improve the uncertainty, several researchers
have integrated fuzzy logic with AHP. Buckley (1985) extended Saaty's AHP to the case
where the decision makers are allowed to use fuzzy ratios in place of exact ratios to handle
the difficulty of people to assign exact ratios when comparing two criteria and derive the
fuzzy weights of criteria by geometric mean method. In this study, we employ Buckley's
method to fuzzify hierarchical analysis by employing fuzzy numbers in the pairwise
comparisons and to find the fuzzy weights. This has led to the term called Fuzzy Analytic
Hierarchy Process (Fuzzy AHP or FAHP). A brief summary of Fuzzy logic is presented in
the next section.

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2.7 Fuzzy Logic
The concept of fuzzy sets forms the basis of fuzzy logic. It was introduced by Zadeh (1965)
as an extension of the set theory. A classical, or a crisp set, is one which assigns grades of
membership of either 0 or 1 to objects within their universe of discourse. In other words,
objects either belong to or do not belong to a certain class; or object either possesses a certain
property, or they do not; there is no middle ground. A fuzzy set, on the other hand, is one
which assigns grades of membership between 0 and 1 to objects within its universe of
discourse. If X is a universal set whose elements are {x}, then, a fuzzy set A is defined by, its
membership function,
µA: X  [0, 1]
which assigns to every member x a degree of membership µA in the interval [0, 1]
(Simonovic, 2001). Numbers based on these sets are known as fuzzy numbers. The most
common types are the triangular and trapezoidal fuzzy numbers. Other types of fuzzy
numbers are also possible, such as bell-shaped or Gaussian fuzzy numbers, as well as a
variety of one-sided fuzzy numbers. Triangular fuzzy numbers are defined by three
parameters, while trapezoidal require four parameters. The triangular numbers are used in this
work.

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CHAPTER THREE
METHODOLOGY
3.1 Background
The situation is a decision involving the selection of the most appropriate method of
generating electricity in an industrial facility with a power requirement in megawatts
(>1MW) in the presence of multiple decision criteria. The situation is characterised by the
presence of a number of available alternatives to choose from and more than one decision
criterion. In essence, the decision to be made is that of choosing the „best‟ electricity
generation method from a list of available alternatives while taking into consideration a
number of criteria.
The term „self-generation‟ refers to the in-house generation of electricity by individuals or
commercial bodies for their own activities as against acquiring power from the public grid. It
may also be referred to as own-generation. A finite number of feasible self-generation
methods were evaluated in this work. Feasible methods were considered in the sense that
methods which could not be localized in the city of Ibadan due to some reasons. Such reasons
include the non-availability of the required technology in the country (as is the case for
geothermal), location constraints (e.g. coal plants, offshore wind turbines can only be sited
near coal deposits and offshore waters respectively) and legislative barriers (as is the case for
nuclear plants). Note that all the methods are considered as stand-alone systems without any
support from the public grid. The option of total or partial reliance on the public grid was not
considered because a process manufacturing setup was being considered. Such setups do not
allow for any stoppage due to power failure or any other reason(s). Also, the frequency of
load shedding is mostly erratic and cannot be adequately simulated within the scope of this
study.

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3.2 Theoretical Framework
The VIKOR method is essentially based on the Compromise Programming algorithm. As
earlier shown, the general formulation of a CP approach is expressed as follows:
3.1
1 ≤ p ≤ ∞; j = 1, 2,…, J
Where is the parameter distance of alternative Aj from the ideal
is the value for criterion ci in alternative Aj.
is the ideal value for criterion ci
is the anti-ideal value for criterion ci
wi is the weight of criterion ci,
p is the parameter reflecting the decision-makers‟ concern with respect to the maximal
deviation.
Having determined the distance of different alternatives from the ideal, the compromise
solution, Acp, is obtained from the minimum solution of the optimization of the Lp,j metric
after appropriate values must have been assigned to p and wi.
3.3 Model Description
A brief description of the model is as follows:
3.3.1 Objective
This is to determine the self-generation method which is „closest‟ to the ideal. The ideal can
be defined as that infeasible method which has the best values for all the criteria employed.

33
3.3.2 Alternatives
These refer to the various alternative methods of self-generation of electricity that are
considered. These methods have various values with respect to each criterion function. For
example, alternative Aj has a value of fij with respect to criterion ci.
3.3.3 Criteria
These refer to relevant characteristics, factors and indices of the alternative self-generation
methods. They provide the means for evaluating the attainment level of an objective. Each
criterion ci has a weight of importance, wi associated with it.
3.3.4 Constraints
These are restrictions on attributes and decision variables that can or cannot be expressed
mathematically. Most constraints of the compromise ranking method are of the latter kind.
Some of the constraints of the model are as follows:
 The values of the attributes are all non-negative i.e. , ≥ 0
 The weights of the various attributes take values between 0 and 1, i.e.
 The weights add up to 1, i.e.
The model is expressed mathematically as follows:
3.2
1 ≤ p ≤ ∞; j = 1, 2, …, m; i = 1, 2, …, n
Subject to , ≥ 0,
;

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3.4 Determination of Criteria Weights
This is a critical aspect of the entire decision making process. The various criteria to be
considered have to be assigned weights of importance before the VIKOR ranking
methodology can be adopted. Since the decision/evaluation criteria have varying importance
and meanings, it cannot be assumed that each criterion is of equal importance. There are
many methods that can be employed in the determination of weights. These include the
eigenvector method, weighted least square method, entropy method, AHP (Analytic
Hierarchy Process), and LINMAP (Linear programming techniques for Multidimensional
Analysis Preference) (Chen et al, 2008). The selection of a method depends on the nature and
complexity of the problem. In this work, the Fuzzy Analytic Hierarchy Process (AHP) is the
adopted method of criteria weight determination in order to take into consideration the
inconsistency of human perception and decision making.
The Analytic Hierarchy Process (AHP) constructs a ratio scale associated with the priorities
for the various items compared. In its initial formulation, the AHP is based on the use of pair-
wise comparisons, which lead to the elaboration of a ratio scale. The Fuzzy AHP (FAHP) is
an amendment of the AHP which introduces triangular fuzzy numbers into the pair-wise
comparisons. In order to adequately understand the FAHP method, a brief summary of fuzzy
mathematics is given next.
3.4.1 Fuzzy Arithmetic
Considering two triangular fuzzy numbers A1 l1, m1, u1 and A2 l2, m2, u2, the following
are the laws which govern fuzzy operations (Sun, 2011):
 Addition of fuzzy numbers, 
A1 l1, m1, u1 l2, m2, u2 l1+l2, m1+m2, u1+u2    3.3

35
 Subtraction of fuzzy numbers, 
A1 l1, m1, u1 l2, m2, u2 l1  u2, m1m2, u1 l2   3.4
 Multiplication of fuzzy numbers, 
A1 l1, m1, u1l2, m2, u2 l1l2, m1m2, u1u2    3.5
 Division of fuzzy numbers, 
A1 l1, m1, u1l2, m2, u2l1/u2, m1m2, u1/l2    3.6
 Reciprocal of a fuzzy number
A1
-1
= l1, m1, u1
= 1/u1, 1/m1, 1/l1     3.7
Where l1, m1, u1, l2, m2, u2 > 0
3.4.2 Fuzzy Analytic Hierarchy Process (FAHP)
Furthermore, the steps involved in the FAHP are as follows (Sun, 2011):
1. Construct pair-wise comparison matrices among all the elements/criteria in the dimensions
of the hierarchy system. In this work, 3 Decision Makers are involved in the decision
process. Assign linguistic terms to the pair-wise comparisons by asking which is the more
important of each two dimensions, as following matrix M:
M = =
where aij = {9-1
, 8-1
, 7-1
, 6-1
, 5-1
, 4-1
, 3-1
, 2-1
, 1-1
, 1, 2, 3, 4, 5, 6, 7, 8, 9 for i ≠ j and 1for i = j }
Note that aij = 1/aji
where aij is fuzzy comparison value of criterion i with criterion j. In essence, it gives a
numerical value of the degree of importance of criterion i over j, i.e. it answers the question:
how important is criterion i with respect to criterion j? A fuzzy average of their assessments
makes up the matrix. The meanings of the numbers (linguistic terms) above are shown in
Table 3.1.

36
2. We need to know the vector W = [w1, w2… wn] which indicates the weight of each
criterion. To recover the vector W from the matrix M, a two-step procedure is outlined as
follows:
 Compute the fuzzy geometric mean, ri and fuzzy weight, wi of each criterion using
the following expressions:
ri = (ai1 aijaim) 3.8
wi = ri r1 ri rn
       3.9
Here, ri is the geometric mean of the fuzzy comparison values of criterion i to every other
criterion and itself. wi is the fuzzy weight of the i-th criterion. Each fuzzy parameter is
represented by a triangular fuzzy number. For instance, wi = (lwi, mwi, uwi) where lwi, mwi
and uwi stand for the lower, middle and upper values of the fuzzy weight of the i-th criterion
respectively.
Table 3.1: Membership function of linguistic scale (Sun, 2011)
Fuzzy Number Linguistic Scale of Fuzzy Number
9 Perfect (8,9,10)
8 Absolute (7,8,9)
7 Very good (6,7,8)
6 Fairly good (5,6,7)
5 Good (4,5,6)
4 Preferable (3,4,5)
3 Not Bad (2,3,4)
2 Weak advantage (1,2,3)
1 Equal (1,1,1)
Since the fuzzy weights obtained cannot be employed in the VIKOR computations, the
triangular fuzzy numbers are converted to non-fuzzy numbers. The Centre of Area
(COA) method is employed to compute the Best Non-fuzzy Performance (BNP) value

37
of the fuzzy weights of each criterion. The calculation is done using the following
equation:
BNPwi = [(uwi - lwi) + (mwi - lwi)] / 3 + lwi 3.10
The BNP value of each criterion weight is then used in the VIKOR computations.
3.5 The VIKOR Method
The compromise ranking algorithm VIKOR is based on the above model and has the
following steps (Opricovic & Tzeng, 2004):
(a) Determine the best or ideal value, fi
*
and the worst or anti-ideal value, fi
-
of all criterion
functions, i = 1, 2. . . n;
fi
*
= fij if the i-th function represents a benefit or fij if the i-th function represents a
cost.
fi
-
= fij if the i-th function represents a cost or fij if the i-th function represents a
benefit. The alternative which has all its criteria values as being ideal is known as the utopian
alternative. The utopian does not exist.
(b) Compute the values of the utility measure, Sj and the regret measure, Rj for the alternative
j, using the following expressions:

38
Where wi are the weights of criteria which express the DM‟s evaluation of the relative
importance of the criteria.
(c) Compute the values of Qj for each alternative j using equation 3.13:
Qj = 3.13
where = , = , = , = ; and is introduced as a weight
for the strategy of maximum group utility whereas is the weight of the individual
regret.
(d) Rank the alternatives, sorting by the values S, R and Q in decreasing order. The results are
three ranking lists.
(e) Propose as the compromise solution the alternative A(1)
which is the best ranked by the
measure Q (i.e. the minimum) if the following two conditions are satisfied:
1. Acceptable advantage: Q(A(2)
) – Q(A(1)
) ≥ DQ 3.14
where A(2)
is the alternative with the second best position in the ranking list by Q;
3.15
where m is the total number of alternatives being evaluated.
1. Acceptable stability in decision making: The alternative A(1)
must also be the best
ranked by S or/and R. This compromise solution is stable within a decision making
process, which could be the strategy of maximum group utility (when > 0.5 is
needed), or “by consensus” = 0.5, or “with veto” ( < 0.5). Here, is the weight of
decision making strategy of maximum group utility.

39
If one of the conditions is not satisfied, then a set of compromise solutions is proposed, which
consists of
 Alternatives A(1)
and A(2)
if only condition 2 is not satisfied, or
 Alternatives A(1)
, A(2)
, . . . , A(M)
if the condition 1 is not satisfied; A(M)
is determined
by the relation:
Q(A(M)
) - Q(A(1)
) < DQ for maximum M (the positions of these alternatives are „„in
closeness‟‟).
The VIKOR method is an effective MCDM tool, specifically applicable to those situations
where the decision maker is not able, or does not know how to express his/her preference at
the beginning of the decision-making process (Opricovic & Tzeng, 2004). The resulting
compromise solution can be accepted by the decision maker because it provides a maximum
group utility of the „majority‟ and a minimum individual regret of the „opponent‟. The
compromise solutions can be the base for negotiations, involving the decision maker‟s
preference on criteria weights. The VIKOR results depend on the ideal solution, which stands
only for the given set of alternatives. Inclusion (or exclusion) of an alternative can affect the
VIKOR ranking of the new set of alternatives.
3.6 Summary of the Procedure
The selection procedure can be summarised as follows:
1. Select the feasible alternatives for the decision problem.
2. Select the relevant decision criteria by which the selected alternatives will be evaluated.
3. Determine the weights of importance of the selected criteria.
4. Determine the best and worst values of each decision criterion.
5. Using the VIKOR equations 3.11 and 3.12, determine the Sj and Rj values of each
alternative.

40
6. Using the best and worst values of Sj and Rj, determine the Qj values of each alternative for
varying values of the weight for the strategy of maximum group utility, .
7. Rank the alternatives in ascending order of Sj, Rj and Qj values.
8. Select the compromise alternative or set of alternatives based on the conditions of the
VIKOR model

41
False
True
False
True
Figure 3.1: Flowchart of the VIKOR Method
Start
Obtain values of fij, wi; for
i = 1, 2. . . n; j=1,…,m
Determine the best fi
*
and the worst fi
-
values of all criterion functions
Compute Sj and Rj for each alternative
Determine ,
Choose a value for v and compute Qj for
each alternative
Rank alternatives in ascending order of
Sj, Rj and Qj values
Compute DQ
Q(A(2)
) –
Q(A(1)
) ≥ DQ
Q(A(1)
) = S(A(1)
) or
Q(A(1)
) = R(A(1)
)
Determine maximum M for
Q(A(M)
) - Q(A(1)
) < DQ
Propose A(1)
as the compromise solution
Propose A(1)
,…, A(M)
as the
compromise solutions
Propose A(1)
and A(2)
as the compromise
solutions
End

42
CHAPTER FOUR
DATA COLLECTION, ANALYSIS AND APPLICATION
4.1 Selection of Criteria
The first step in multicriteria optimization is the establishment of criteria for system
evaluation (Opricovic et al, 2004). These criteria, as the name suggests, must at least be two
in order to qualify to be multicriteria. However, these different criteria conflict with one
another. This means that, in a particular alternative, the improvement of one criterion may
lead to the deterioration of one or more other criteria. In reaching a decision, the various
criteria are assigned numerical weights of importance to show their priority.
In deciding upon a particular means of power generation, a number of factors are usually
considered. In this study, 5 relevant criteria are considered. They are
1. Setup cost
2. Annual operation and maintenance costs
3. amount of pollution from exhaust fumes
4. Noise
5. Capacity factor
4.1.1 Setup Cost
The setup or first cost comprises the costs incurred in purchasing and installing the facility.
This is an important factor because it may determine the company‟s ability to purchase the
facility or not. The cost may be so high as to deter the desire to purchase. The initial setup
cost also comprises other costs such as shipment costs, taxes and the cost of housing the
facility and fuel storage. The desired objective with respect to this factor is to keep it at a
minimum.

43
4.1.2 Annual Maintenance and Operation Costs
This is one of the more important decision criteria in this study. This is due to it being an
addition to the annual expenditure of the company for the entire lifespan of the facility. As a
result, it is essential that it be kept at a reasonable low. It comprises the following costs:
 Costs of fuel to keep the facility running
 Costs of replacement parts and other preventive and corrective maintenance activities
 Wages of operators and maintenance crew
4.1.3 Pollution from Exhaust Fumes
In recent times, there have been several calls for the reduction in emission of greenhouse
gases (GHGs) such as carbon (IV) oxide, methane and nitrogen (IV) oxide. These have
resulted in various national and international treaties and laws to curb activities that produce
such emissions. One of such activities is power generation. Almost all industrial generators in
the country use hydrocarbon-based fuels. As a result, their exhaust fumes comprise
significant amounts of GHGs. These fumes are also harmful to the persons and households in
surrounding areas. Hence, a reduction in their amounts is of some importance to
environmental sustainability and occupational health and safety.
Most reputable organizations have a blueprint for a gradual reduction in their emissions and
hence limiting their „carbon footprint‟ in the environment. This study will take into
consideration the emission of carbon (IV) oxides only. This is because it is by far the most
common emission from sources that utilize hydrocarbon fuels.
4.1.4 Noise
There is always some noise that accompanies any mechanical system that comprises moving
parts. Noise may also be due to vibration or parts in contact. The negative impacts of
excessive noise on occupational health, hearing in particularly, make noise levels an

44
important decision criterion in this study. Studies have shown that continuous exposure to
noise levels above 95 dB may result in partial deafness.
In this study, noise level is measured at 1m from the facility. Its unit is decibel (dB).
4.1.5. Capacity Factor
The net capacity factor of a power plant is the ratio of the actual output of the plant over a
period of time and its potential output if it had operated at full capacity the entire time. The
capacity factor of a facility may depend on various factors. The ratio may be low due to long
hours of downtime as a result of breakdowns, routine and non-routine maintenance. Also, the
availability of the required fuel may affect the ratio. This is especially true for renewable
energy technologies such as wind and solar. Their fuels, wind and sunlight that is, are not
available or adequate at all times. The result of this is their under-utilization.
4.2 Identification of Alternatives
This study analyzes a total of 8 power sources with regard to the 5 criteria mentioned above.
The alternatives are those that are feasible in a typical industrial estate in the country. As a
result, geothermal, tidal, offshore wind and nuclear sources are not considered. Factors such
as location constraints, unavailability of the technology in the country as well as government
legislation may affect the utilization of these technologies by private or commercial entities
in Nigeria at present. The 8 alternatives being considered are as follows:
1. Diesel Internal Combustion Generators
2. Natural Gas Internal Combustion Generators
3. Bi-fuel or Dual Fuel Internal Combustion Generators
4. Microturbine
5. Solar PV Cells

45
6. Wind Turbine
7. Gas Turbine Plant
8. Fuel Cell
4.3 Determination of Criteria Weights
The method to be used is the Fuzzy Analytic Hierarchy Process (AHP). The steps in the
method are as follows:
1. Construct a matrix from pair-wise comparisons among all the criteria to determine
their relative importance. This is done by assigning linguistic terms to the pair-wise
comparisons by asking which is the more important of each two dimensions. The
fuzzy triangular numbers from Table 3.1 were used. The matrix is as follows:
M =
where aij is the fuzzy comparison value of criterion i with criterion j as discussed in
Section 3.3.2.
The pair-wise comparison was done by 3 independent decision makers (DMs) in order to get
the final weights. Each DM gives a rating to each alternative with regards to the criteria. The
preference ratings are given on the scale of fuzzy numbers provided in Table 3.1. The pair-
wise ratings by each DM are given in Tables 4.1, 4.2 and 4.3.
For instance, a rating of (1, 1, 1) which means „equal importance‟ was assigned to the
comparison of capital cost with itself by all 3 DMs. Also, the comparison of annual O & M
cost with capital cost was given a rating of (6, 7, 8) by DM 1. From Table 3.1, this signifies a
„very good‟ importance rating; i.e. annual O&M cost has a very good importance when

46
compared to capital cost. DM 2 and 3 gave a rating of and ) which
respectively mean absolute and perfect importance.
Table 4.1: Decision Maker 1's Pair-wise Ratings
Capital Cost Annual O&M
Cost
Emission Noise Capacity Ratio
Capital Cost (1, 1, 1) (2, 3, 4) (4, 5, 6)
Annual O&M
Cost
(6, 7, 8) (1, 1, 1) (4, 5, 6) (8, 9, 10)
Emission (1, 1, 1) (4, 5, 6)
Noise (1, 1, 1)
Capacity Ratio (8, 9, 10) (4, 5, 6) (4, 5, 6) (8, 9, 10) (1, 1, 1)
Cost
Capital Cost (1, 1, 1)  
Annual O&M
Cost
 (1, 1, 1)  
Emission (1, 1, 1) 
Noise (1, 1, 1)
Capacity Ratio     (1, 1, 1)
Cost
Capital Cost (1, 1, 1)  )
Annual O&M
Cost
) (1, 1, 1) ) )
Emission (1, 1, 1) )
Noise (1, 1, 1)
Capacity Ratio ) ) ) ) (1, 1, 1)

47
The empty spaces can be evaluated from their respective inverses using equation 3.7. For
instance, the comparison rating of capital cost with capacity ratio can be determined from that
of capacity ratio with capital cost.
In order to obtain the matrix M, the computations are done with fuzzy arithmetic and a full
account is given in Appendix II. However, a brief illustration is given below using equation
4.1:
aij = (aij
1
aij
2
aij
3
)1/3
4.1
where aij
k
is the fuzzy comparison rating of the i-th criterion with the j-th criterion by the k-th
decision maker.
As expected, a value of (1, 1, 1) is given to the comparison of capital cost with itself. The
comparison rating of annual O&M cost with capital cost is determined by the fuzzy
multiplication operation as illustrated in equation 3.5.
a21 = [(6, 7, 8) )] 1/3
= (6.96, 7.96, 8.96)
Similarly, the other values were determined. Some others were determined using the fuzzy
inverse operation as illustrated in equation 3.7. For instance, the comparison rating of capital
cost with annual O&M cost was determined from the inverse of that of annual O&M cost
with capital cost. This is illustrated as follows:
a12 = 1 / a21 = (6.96, 7.96, 8.96)-1
= (8.96-1
, 7.96-1
, 6.96-1
) = (0.11, 0.13, 0.14)

48
Consequently, the matrix M looks as shown in Table 4.4.
Table 4.4: Matrix M
Cost
Capital
Cost
Annual
O&M Cost
Emission
Noise
Capacity
Ratio
2. The fuzzy geometric mean and fuzzy weights of each criterion are then respectively
calculated using Equations 3.8 and 3.9 respectively. The fuzzy computations are
shown in the Appendix. A brief illustration of the computation is shown below:
r1 = (a11 a12a13 a14a15)1/5
= [(1, 1, 1) (0.11, 0.13, 0.14) (1.59, 2.62, 3.63) (3.63,
4.72, 5.77) (0.11, 0.12, 0.14)] = [(1×0.11×1.59×3.63×0.11)1/5
, (1×0.13×2.62×4.72×0.12)1/5
,
(1×0.14×3.63×5.77×0.14)1/5
] = (0.587, 0.720, 0.837)
w1 = r1 r1 r2r3  r4r5
(0.587, 0.720, 0.837) (0.587, 0.720, 0.837) (1.883,
2.183, 2.526) (0.498, 0.608, 0.765) (0.200, 0.231, 0.266) (4.042, 4.621,
5.175)
(0.587,0.720,0.837)




The remaining values were similarly computed.

49
Table 4.5: Fuzzy Geometric Means and Criteria Weights
i 1 2 3 4 5
ri (0.587, 0.720, 0.837) (1.883, 2.183, 2.526) (0.498, 0.608, 0.765) (0.200, 0.231, 0.266) (4.042, 4.621, 5.175)
wi     
The Centre of Area (COA) method is applied to compute the Best Non-fuzzy Performance
(BNP) value of the fuzzy weights of each criterion. The computations were done using
Equation 3.10 and are shown in Appendix II. An illustration is given below to show the
computation for the first criterion i.e. capital or setup cost. The results are summarized in
Table 4.6.
BNPw1 = [(uw1 - lw1) + (mw1 - lw1)] / 3 + lw1 = [(0.116 – 0.062) + (0.086 – 0.062)] / 3
+ 0.062 = 0.085
where lwi, mwi and uwi stand for the lower, middle and upper values of the fuzzy weight of
the i-th criterion respectively.
Table 4.6: Best Non-Fuzzy Performance Values of the Criteria
w1 w2 w3 w4 w5
BNPwi 0.085 0.269 0.072 0.024 0.550
The BNP values are thus the weights to be used in the VIKOR computations.

50
4.4 VIKOR Computations
The values of the alternatives with respect to the different criteria are as shown in Table 4.2.
These values were obtained from various sources which include selected local companies,
manufacturers‟ manuals and data from foreign manufacturers and users. A more detailed
analysis of these sources is given in Appendix I.
Table 4.7: Values of the Alternatives with respect to each Criterion
j Alternative Capital Cost
(N/kW)
Annual O&M
Cost (N mil)
Emission
(g/kWh)
Noise
(dB)
Capacity
Ratio (%)
1 Diesel I.C.E 40,000 190.9 778 110 85
2 Gas I.C.E 80,000 73.8 654 85 90
3 Bi-fuel I.C.E 50,000 106.8 691 90 95
4 Microturbine 184,800 106.2 624.6 73 95
5 Solar-PV 798,140 8.0 0 10 20
6 Wind 202,944 4.1 0 90 40
7 Gas Turbine 252,000 66.8 443 90 90
8 Fuel Cell 759,360 89.9 30 60 95
The capital costs are computed per nameplate kilowatt i.e. the manufacturer‟s rating. The
capital, operation and maintenance costs of the diesel and gas combustion engines were
obtained from local suppliers and companies who use them. The costs of others were
computed with an additional factor of 20% from manufacturer‟s manuals and studies done
overseas. The factor is required to take into consideration additional expenses in localizing
the technologies in the country. Such expenses include shipping costs and import duties. Fuel
costs are calculated as N110 per litre for diesel and N35 per cubic metre for natural gas. All
alternatives are assumed to run on natural gas except the diesel engine and the renewable

51
energy technologies. The bi-fuel alternative runs on 30% diesel and 70% natural gas. All
costs are based on facilities rated at 1 MW. It has been observed that the capital cost vary
with the rating, usually reducing per KW as the rating increases (See Appendix I for more
details).
The CO2 emissions were obtained from manufacturer‟s manuals and a meta-analysis study by
Benjamin Sovacool (Sovacool, 2008). With regards to wind and solar-PV, only emissions at
source were considered. Noise levels are measured at a distance of 1 metre from the facility.
Measurement values are from manufacturer‟s manuals.
The next step in the VIKOR procedure is to determine the best, fi
*
and worst values, fi
-
of the
criteria. In the case of capacity ratio, the highest is best and the lowest is worst. This is
because the objective is to get the alternative with the highest possible ratio. However, in the
case of the other criteria, it is vice versa. This is because the objective is to minimize them as
much as possible in the alternative to be selected. The best and worst values are shown in
Table 4.8.
Figure 4.8: Best and Worst Values of each Criterion Function
Capital Cost
(N/kW)
Annual O&M
Cost (N mil)
Emission
(g/kWh)
Noise (dB) Capacity
Ratio (%)
fi
* 40,000 4.1 0 10 95
fi
- 798,140 190.9 778 110 20
The values of the utility measure, Sj and the regret measure, Rj were determined for each
alternative using equations 3.11 and 3.12 of the VIKOR model and are shown in Table 4.9.

52
Figure 4.9: Sj and Rj Values of the Alternatives
j 1 2 3 4 5 6 7 8
Sj 0.438 0.220 0.232 0.236 0.556 0.441 0.211 0.244
Rj 0.269 0.100 0.148 0.147 0.550 0.403 0.090 0.124
Using the data in Table 4.9, the values of Qj are computed with equation 3.13 using 3
different values of as a means of Sensitivity Analysis. An illustration is given below.
From Equation 3.13,
= minj = = 0.211, = maxj = = 0.556, = minj = = 0.090 and =
maxj = = 0.550
For = 0, 0(0.438–0.201)/(0.556–0.211) + (1–0)(0.269–0.090)/(0.550–0.090) = 0.389
For 0.5, 0.5(0.438–0.201)/(0.556–0.211) + (1–0.5)(0.269–0.090)/(0.550–0.090) =
0.524
For 1, 1(0.438–0.201)/(0.556–0.211) + (1–1)(0.269–0.090)/(0.550–0.090) = 0.658
The 3 different values of are utilized for sensitivity analysis. When the strategy of
maximum group utility is adopted and the individual regret ignored, ( = 1) can be selected
for the calculation, whereas when the individual regret is considered and the strategy of
maximum group utility ignored, ( = 0) can be selected. Generally speaking, when decision
makers are both concerned about the strategies of maximum group utility and the minimum
individual regret, then = 0.5 is to be selected. This selection is decided based on the

53
preference of the decision makers. The values of Q and the respective rankings (in brackets)
are as follows:
Figure 4.10: Qj values of the Alternatives (Rankings are in brackets)
Alternatives
Diesel
I.C.E.(1)
Gas
I.C.E.(2)
Bi-Fuel
I.C.E.(3)
Microturbine
(4)
Solar-
PV(5)
Wind(6) Gas
Turbine(7)
Fuel
Cell (8)
Qj
= 0 0.389 (6) 0.022 (2) 0.126 (5) 0.124 (4) 1.00(8) 0.680 (7) 0.000 (1) 0.074 (3)
= 0.5 0.524 (6) 0.024 (2) 0.094 (4) 0.098 (5) 1.00(8) 0.674 (7) 0.000 (1) 0.085 (3)
= 1 0.658 (6) 0.026 (2) 0.061 (3) 0.072 (4) 1.00(8) 0.667 (7) 0.000 (1) 0.096 (5)
4.5 Results and Discussion
A summary of the rankings in Section 4.4 are displayed in Table 4.11.
The VIKOR algorithm stipulates two conditions to be met in order to choose the best
compromise alternative. These conditions have been mentioned in Section 3.5 (equations
3.14 and 3.15).
With regards to the three Qj rankings, none satisfied the first condition of acceptable
advantage. However, the second condition (acceptable stability in decision making) was met
by all three rankings. In such a case, the compromise solution is a set of alternatives ranked
A(1)
, A(2)
, . . . , A(M)
where A(M)
is determined by the relation:
Q(A(M)
) - Q(A(1)
) < DQ for maximum M (DQ was determined as ≈ 0.143 from equation 3.15)
For all three Q rankings, the compromise set comprises alternatives 2, 3, 4, 7 and 8. These
correspond to the gas internal combustion engine, bi-fuel engine, microturbine, gas turbine
and the fuel cell.

54
Figure 4.11: Sj, Rj and Qj Rankings of the Alternatives
S Ranking R Ranking Q (v = 0)
Ranking
Q (v = 0.5)
Ranking
Q (v = 1)
Ranking
1st
Gas Turbine Gas Turbine Gas Turbine Gas Turbine Gas Turbine
2nd
Gas I.C.E. Gas I.C.E. Gas I.C.E. Gas I.C.E. Gas I.C.E.
3rd
Bi-Fuel I.C.E. Fuel Cell Fuel Cell Fuel Cell Bi-Fuel
I.C.E.
4th
Microturbine Microturbine Microturbine Bi-Fuel I.C.E. Microturbine
5th
Fuel Cell Bi-Fuel I.C.E. Bi-Fuel I.C.E. Microturbine Fuel Cell
6th
Diesel I.C.E. Diesel I.C.E. Diesel I.C.E. Diesel I.C.E. Diesel I.C.E.
7th
Wind Wind Wind Wind Wind
8th Solar-PV Solar-PV Solar-PV Solar-PV Solar-PV
The rankings in Table 4.11 reflect the perception that changes in strategies (i.e. value of v) of
decision makers may affect the outcome of the rankings to a certain degree. It is clear that
most alternatives maintain similar relative rankings under different strategies.
Table 4.12: Compromise Set of Alternatives
Gas Internal Combustion Engine
Bi-fuel Internal Combustion Engine
Microturbine
Gas Turbine
Fuel cell

55
The analysis of the results shows that the same alternatives appear in the compromise set of
all three scenarios. These alternatives can be said to provide the best compromise or balance
of all criteria considered. As a result, one or a combination of any of these alternatives will
serve as a viable option for generating power for industrial setups in the country. The solar-
PV option consistently ranked least in all three Q rankings. A major contributory factor to
this is also its low capacity factor. With further research and advancements, it will become
cheaper, more efficient in utilizing and storing solar energy and, hence, more readily
applicable for industrial uses. The same is the case with the wind turbine technology. Lower
capital costs and higher availability will improve its chances of commercialization in little or
no time.
It must be reiterated that the choice of the DM should be from the compromise set as the
elements of this set give the best balance of the 5 criteria considered. The choice may be one
or any combination of alternatives which belong to the compromise set depending on a
number of factors. These factors may include the budget available to the decision maker,
availability of the alternatives and the ease of their maintenance, availability of spare parts,
and so on. All these factors will ultimately affect the decision maker‟s final decision.
Without doubt, the weights of the decision criteria have played a very important role in the
rankings and the consequent decision making process. Therefore, the method of determining
their weight is one which must, as much as possible, be devoid of any bias or error. The
Fuzzy AHP method is one such method. However, the decision making process shall remain
subjective so long as humans are involved. We introduce pre-conceived sentiments and bias
into the process; thus, making it inconsistent and, sometimes, incorrect. A way out is the
employment of a large number of DMs in the process with the hope that the bias of one DM
will be cancelled out by another.

56
CHAPTER FIVE
CONCLUSION AND RECOMMENDATIONS
5.1 Conclusion
The importance of employing the most appropriate means of electricity generation by a
commercial or profit-oriented organisation in an economy like Nigeria‟s cannot be over-
emphasized. This is so because the selected means of generation has a huge impact on the
organisation‟s financial books and, hence, their ability to break even. Consequently, the
decision of selecting an appropriate method is too important to be handled with levity. The
decision must consider several conflicting objectives such as technological, environmental,
economic, health, etc. The VIKOR method has been observed to be very useful in those
particular cases wherein there are multiple conflicting variables to consider.
The importance of the criteria was evaluated by 3 independent decision makers (DMs) or
experts. In order to reduce the bias and error due to the subjective judgements of the DMs,
fuzzy logic was incorporated in the Analytic Hierarchy Process (AHP) to form a method
aptly called Fuzzy Analytic Hierarchy Process (FAHP).
The VIKOR method focuses on ranking and selecting the best from a finite set of alternatives
in the presence of multiple conflicting criteria. It determines a compromise solution that
could be accepted by the decision makers because it provides a maximum group utility for
the „„majority‟‟, and a minimum of individual regret for the „„opponent‟‟. This research has
utilized a Fuzzy AHP and VIKOR model to evaluate various means of self-generation of
power for a large facility. Eight generation methods were selected and evaluated while taking
five decision criteria into consideration. The criteria considered were the setup (capital) cost,
annual operation and maintenance costs, amount of greenhouse emissions (more specifically,
CO2), noise and the alternative‟s capacity factor. Using 3 decision makers, the Fuzzy AHP

57
method determined that the capacity factor carried the most weight out of all 5 criteria. The
results of the multi-criteria analysis suggested that the gas internal combustion engine, bi-fuel
engine, microturbine, gas turbine and the fuel cell form the most satisfactory means of power
generation, especially for a large industrial facility with a power demand above 1MW.
5.2 Recommendations
Some of the concepts outlined in this work may open up exciting new research paths in multi-
criteria decision making. Others may require modifications in order to make them more
accurate and their results more realistic. Huang et al (2009) suggested a revised form of the
VIKOR method in which the perspective of regret theory is employed. In the proposed
model, two different kinds of regret, namely the discontent and choice less utilities, are
included to reflect the choice behaviour of decision makers.
The VIKOR method is designed in such a way as to be sensitive to criteria weights (wi). As a
result, further work may be done using the method to test the results with alternative weights
as a form of sensitivity analysis. Another means of improving the integrity of results is to
engage a larger number of decision makers in the evaluation and assignment of criteria
weights. Also the weight v has an important role in identifying the ranking. Of course, the use
of other MCDM tools like TOPSIS, ELECTRE II, etc in comparing results of the decision
making process will be beneficial. The comparison of VIKOR and TOPSIS has been done by
Opricovic and Tzeng (2004).

58
REFERENCES
Abrishamchi, A., Ebrahimian, A., Tajrishi, M., & Mariño, M. A. (2005). Case Study:
Application of Multicriteria Decision Making to Urban Water Supply. Journal of
Water Resources Planning and Management, 131, 326-335.
Adenikinju, A. (2005). Analysis of the cost of infrastructure failures in a developing
economy: The case of the electricity sector in Nigeria. Nairobi: African Economic
Research Consortium.
Adeyeye, A. D., & Oyawale, F. A. (2010). Multi-objective methods for welding flux
performance optimization. RMZ – Materials and Geoenvironment, 57, 251–270.
Athawale, V. M., Chatterjee, P., & Chakraborty, S. (2010). Selection of Industrial Robots
using Compromise Ranking Method. 2010 International Conference on Industrial
Engineering and Operations Management, (pp. 1-5). Dhaka, Bangladesh.
Ayodele, A. S. (2003). Improving and Sustaining Power (Electricity) Supply for
Socio-Economic Development in Nigeria. Ibadan: NISER.
Buckley, J. (1985). Fuzzy hierarchical analysis. Fuzzy Sets and Systems, 17, 233–247.
Caves, D., Herriges, J., & Windle, R. (1992). The cost of electric power interruptions in the
industrial sector: Estimates derived from interruptible service programmes. Land
Economics, 68, 49–61.
Cetin, M. K., & Cetin, E. I. (2010). Multi-Criteria Analysis of Banks‟ Performances.
International Journal of Economics and Finance Studies, 2, 73-78.
Chang, C.-L., & Hsu, C.-H. (2009). Multi-criteria analysis via the VIKOR method for
prioritizing land-use restraint strategies in the Tseng-Wen reservoir watershed.
Journal of Environmental Management, 2, 1-5.
Chen, Y.-C., Lien, H., Tzeng, G.-H., & Yang, L.-S. (2008). Fuzzy MCDM approach for
selecting the best environment-watershed plan. Journal of Environmental
Management, 1-13.
Cristóbal, J. R., Biezma, M. V., Martínez, R., & Somoza, R. (2009). Selection of Materials
under Aggressive Environments: The Vikor Method. 3rd International Conference
on Integrity, Reliability and Failure, (pp. 1-8). Porto.
Duckstein, L., & Opricovic, S. (1980). Multiobjective optimization in river basin
development. Water Resources Research, 16, 14–20.
Eisenhardt, K., & Zbaracki, M. (1992). Strategic decision making. Strategic Management
Journal, 17–37.
Foster, V., & Steinbuks, J. (2009). Paying the Price for Unreliable Power Supplies. African
Sustainable Development Front Office, The World Bank.

59
Freimer, M., & Yu, P. L. (1976). Some new results on compromise solutions for group
decision problems. Management Science, 22, 688–693.
Gagné, C., Gravel, M., & Price, W. L. (2005). Using metaheuristic compromise programming
for the solution of multiple-objective scheduling problems. Journal of the
Operational Research Society, 56, 687–698.
Ganoulis, J. (2001). Ranking Alternative Strategies of Agricultural Water Use in the
Euro-Mediterranean Area. Euro-Mediterranean Forum of Economic Institutes,
(pp. 1-17). Marseille.
Huang, J.-J., Tzeng, G.-H., & Liu, H.-H. (2009). A Revised VIKOR Model for Multiple
Criteria Decision Making - The Perspective of Regret Theory. In Y. Shi,
MultiCriteria Decision Making (pp. 761–768). Heidelberg: Springer-Verlag Berlin
Heidelberg.
Hwang, C., & Yoon, K. (1981). Multiple Attribute Decision Making; Methods, Application -
A State-of-the-Art Survey. Berlin–Heidelberg: Springer–Verlag.
Iwayemi, A. (2008). Nigeria‟s Dual Energy Problems: Policy Issues and Challenges.
International Association for Energy Economics, Fourth Quarter, 17-21.
Iyanda, O. (1982). Cost and marketing implications of electric power failures on high-income
households in Lagos. The Nigerian Journal of Economic and Social Studies, 24,
169–84.
Lee, K. S., & Anas, A. (1992). Impacts of Infrastructure Deficiencies on Nigerian
Manufacturing: Private Alternatives and Policy. Washington, D.C: World Bank,
Infrastructure and Urban Development Department.
Lee, K., & Anas, A. (1991). Manufacturers‟ responses to infrastructure deficiencies in
Nigeria: Private alternatives and policy options. Economic Reform in Sub-Saharan
Africa. A World Bank Symposium.
Lipman, T. E., Edwards, J. L., & Kammen, D. M. (2004). Fuel cell system economics:
comparing the costs of generating power with stationary and motor vehicle PEM fuel
cell systems. Energy Policy, 101–125.
Malakooti, B. (1991). A multiple criteria decision making approach for the assembly line
balancing problem. International Journal of Production Research, 29, 1979-2001.
Mintzberg, H., Raisinghani, D., & Theoret, A. (1976). The structure of “unstructured”
decision processes. Administrative Science Quarterly, 246–275.
Nachtnebel, H. P. (1994). Comparison of multi-criterion modelling techniques and guidelines
for selection. In J. J. Bogardi, & H.-P. Nachtnebel, Multicriteria decision analysis in
water resources management (pp. 203-234). Paris: IHP of UNESCO.

60
Opricovic, S. (1998). Multicriteria Optimization of Civil Engineering Systems.
Philadelphia: Faculty of Pennsylvania.
Opricovic, S., & Tzeng, G.-H. (2004). Compromise solution by MCDM methods: A
comparative analysis of VIKOR and TOPSIS. European Journal of Operational
Research, 156, 445–455.
Osunsanya, B. (2008). Meeting Nigeria‟s Power Demand. U.S – Africa Infrastructure
Conference (pp. 1-23). Washington D.C.: Oando PLC.
Saaty, T. L., & Sodenkamp, M. (2010). The Analytic Hierarchy and Analytic Network
Measurement Processes: The Measurement of Intangibles. Decision Making under
Benefits, Opportunities, Costs and Risks. In C. Zopounidis, & P. M. Pardalos,
Handbook of Multicriteria Analysis (pp. 91-166). Heidelberg: Springer.
Sambo, A. S. (2008). Matching Electricity Supply with Demand in Nigeria. International
Association for Energy Economics, Fourth Quarter, 32-36.
Sambo, A. S. (2009). Strategic Developments In Renewable Energy In Nigeria. International
Association for Energy Economics, Fourth Quarter , 15-19.
Simonovic, S. P. (2001). Fuzzy Set Ranking Methods and Multiple Expert Decision Making.
Ontario: University of Western Ontario.
Smit, K. (2002, January 18). DER Equipment: Fuel Cells: Cost. Retrieved August 4, 2011,
from Distributed Energy Resources Guide: Fuel Cells - Cost :
http://www.energy.ca.gov/distgen/equipment/fuel_cells/cost.html
Sovacool, B. K. (2008). Valuing the greenhouse gas emissions from nuclear power: A critical
survey. Energy Policy, 36, 2950.
Stadler, W. (1988). Fundamentals of Multicriteria Optimization: Multicriteria Optimization
in Engineering, in the Science. New York: Plenum Press.
Sun, C.-C. (2011). A hybrid performance evaluation system for notebook computer ODM
companies. African Journal of Business Management, 5, 987-1000.
Tecle, A., Fogel, M., & Duckstein, L. (1988). Multicriterion analysis of forest watershed
management alternatives. Water Resources Bulletin, 1169-1178.
Trincas, G. (2010). Ship Design, A Rational Approach. Trieste: University of Trieste.
Ukpong, I. I. (1973). The economic consequences of electric power failures. The Nigerian
Journal of Economic and Social Studies, 15, 53–74.
WMI. (2010). Operational and Maintenance Costs for Wind Turbines. Retrieved August 8,
2011, from Wind Measurement International:
http://www.windmeasurementinternational.com/wind-turbines/om-turbines.php

61
Yu, P. (1973). A class of solutions for group decision problems. Management Science, 19,
936-946.
Yu, P. L., & Leitmann, G. (1976). Compromise solutions, domination structures and
Salukvadze‟s solution. In G. Leitmann, Multicriteria decision making and differential
games (pp. 85–101). New York: Plenum.
Zadeh, L. (1965). Fuzzy sets. Information Control, 8, 338-353.
Zeleny, M. (1974). A Concept of Compromise Solution and the Method of the Displaced
Ideal. Computer & Operations Research , 479–496.
Zeleny, M. (1973). Compromise programming. In M. Zeleny, & J. L. Cochrane, Multiple
Criteria Decision-Making (pp. 263-301). Columbia: University of South Carolina
Press.
Zeleny, M. (1982). Multiple Criteria Decision Making. New York: McGraw-Hill.

62
APPENDIX
I. Calculation of Costs and other Data in Table 4.1
 Capital Costs: The costs were derived as follows:
1. Diesel I.C.E: N40 million was obtained from CAT-MANTRAC quotation dated
26th
July, 2011 for a 1MVA diesel generator. Hence, a capital cost of N40, 000/kW.
2. Gas I.C.E: A cost of N60 million (twice that of a diesel engine of the same rating)
was given by Generac Power Systems, Inc.
3. Bi-fuel Engine: A cost of N50 million was also obtained from Generac Power
Systems Inc.
4. Microturbine: A cost of $1320/kW was obtained from a report prepared for the
Environmental Protection Agency (Combined Heat and Power Partnership Program),
Washington D.C. dated December 2008. An exchange rate of N140 to a dollar gave
the price of N184, 000/kW.
5. Solar-PV Cell: Based on the US Energy Information Administration (EIA) which
forms the basis for the calculation of 2007 Annual Energy Outlook, the estimated
capital cost of constructing a solar-PV power generating plant was $4751/kW. An
additional factor of 20% was added to take into consideration such costs as shipping
costs, import tariffs, etc. this gave a cost of $5701/kW which corresponds to
N798,140/kW.
6. Wind Turbine: Based on the same source as stated above for solar-PV, the capital
cost is $1449.6/kW after adding the additional factor of 20%. This corresponds to
N202,944/kW.
7. Gas Turbine: Based on the same source as above, the capital cost is $1800/kW
which corresponds to N252,000/kW.

63
8. Fuel Cell: A cost of $5424/kW was quoted by the 2007 Annual Energy Outlook.
This corresponds to N759,360/kW.
 Annual Maintenance and Operation Costs: these costs comprise the maintenance
and fuel costs only. Labour costs are assumed to be equal for all options.
1. Diesel Engine: Maintenance costs amount to N4 million per year. This was
obtained from a Lagos-based firm called Powerworx Nig. Ltd. Fuel costs are
calculated as follows:
Annual Fuel Cost = Unit Diesel cost × Fuel Consumption Rate × 8000 hours (1 year)
= N110/litre × 212.4 litres/hr × 8000 hrs = N186.9 million
Total Annual O & M cost = N190.9 million
2. Gas Engine: Maintenance costs are more than double those of the diesel engine.
They amount to about N10 million. Fuel costs are calculated as follows:
Annual Fuel Cost = Unit Gas cost × Fuel Consumption Rate × 8000 hours (1 year) =
N35/m3
× 228 m3
/hr × 8000 hrs = N63.8 million
3. Bi-fuel Engine: Maintenance costs are about N6 million. For an engine that
operates on 30% diesel and 70% natural gas, the fuel costs are as follows:
Annual Fuel Cost = 0.3(N186.9 million) + 0.7(N63.8 million) = N100.76 million
4. Microturbine: A maintenance cost of $0.02/kWh was given by the California
Energy Commission. This corresponds to N17.92 million. Fuel costs are calculated as
follows:
N35/m3
× 344.8 m3

64
5. Solar-PV Cell: The California Energy Commission states an annual maintenance
cost of 1% of initial capital cost. This is equivalent to N7.98 million. There are no
fuels; hence, no fuel costs.
6. Wind Turbine: An annual maintenance cost of 2% of initial investment is stated
by the same source as above. This amounts to about N4.1 million.
7. Gas Turbine: Maintenance costs may amount to as much as N11.2 million
annually. Fuel costs are as follows:
N35/m3
× 198.67 m3
8. Fuel Cell: A maintenance cost of $0.015/kWh corresponds to N16.8 million
annually (8000 hours). The fuel costs amount to about N73.08 million as calculated
below:
N35/m3
× 261 m3
The calculations above are based on the following assumptions:
1. 1 MMBtu (106
British thermal units) = 28.26 m3
of natural gas at defined temperature
and pressure.
2. Fuel costs are calculated as N110 per litre for diesel and N35 per cubic metre for
natural gas.
3. All costs are based on facilities rated at 1 MW. It has been observed that the capital
cost vary with the rating, usually reducing per KW as the rating increases.
4. All alternatives are assumed to run on natural gas except the diesel engine and the
renewable energy technologies.

65
5. Fuel consumption rates were obtained from CAT‟s (manufacturer‟s) manual.
II. Fuzzy Calculations of the Fuzzy Analytic Hierarchy Process
The computations of the members of the pairwise comparison matrix of the Fuzzy
Analytic Hierarchy Process are as follows:
M =
aij = (aij
1
aij
2
aij
3
)
In the above equation, aij
i
is the fuzzy preference rating of the i-th decision maker.
a11 = (1, 1, 1)
a12 = 1 / a21 = (0.11, 0.13, 0.14)
a13 = [(2, 3, 4) )] 1/3
= (1.59, 2.62, 3.63)
a14 = [(4, 5, 6) )] 1/3
= (3.63, 4.72, 5.77)
a15 = 1 / a51 = (0.11, 0.12, 0.14)
a21 = [(6, 7, 8) )] 1/3
= (6.96, 7.96, 8.96)
a22 = (1, 1, 1)
a23 = [(4, 5, 6) )] 1/3
= (2.88, 3.91, 4.93)
a24 = [(8, 9, 10) )] 1/3
= (6.95, 7.96, 8.96)
a25 = 1 / a52 = (0.17, 0.20, 0.26)
a31 = 1 / a13 = (0.28, 0.38, 0.63)
a32 = 1 / a23 = (0.20, 0.26, 0.35)
a33 = (1, 1, 1)
a34 = [(4, 5, 6) )] 1/3
= (3.91, 4.93, 5.94)

66
a35 = 1 / a53 = (0.14, 0.17, 0.20)
a41 = 1 / a14 = (0.17, 0.21, 0.28)
a42 = 1 / a24 = (0.11, 0.13, 0.14)
a43 = 1 / a34 = (0.17, 0.20, 0.26)
a44 = (1, 1, 1)
a45 = 1 / a54 = (0.10, 0.12, 0.13)
a51 = [(8, 9, 10) )] 1/3
= (7.32, 8.32, 9.32)
a52 = [(4, 5, 6) )] 1/3
= (3.91, 4.93, 5.94)
a53 = [(4, 5, 6) )] 1/3
= (4.93, 5.94, 6.95)
a54 = [(8, 9, 10) )] 1/3
= (7.65, 8.65, 9.65)
a55 = (1, 1, 1)
The computations of the Best Non-fuzzy Performance (BNP) values of the fuzzy weights of
each criterion are as follows:
BNPwi = [(uwi - lwi) + (mwi - lwi)] / 3 + lwi]
Where lwi, mwi and uwi stand for the lower, middle and upper values of the fuzzy weight of
the i-th criterion respectively.
BNPw1 = [(0.116 – 0.062) + (0.086 – 0.062)] / 3 + 0.062 = 0.085
BNPw2 = [(0.351 – 0.198) + (0.262 – 0.198)] / 3 + 0.198 = 0.269
BNPw3 = [(0.106 – 0.052) + (0.073 – 0.052)] / 3 + 0.052 = 0.072
BNPw4 = [(0.037 – 0.021) + (0.028 – 0.021)] / 3 + 0.021 = 0.024
BNPw5 = [(0.719 – 0.424) + (0.555 – 0.424)] / 3 + 0.424 = 0.550

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